Evaluation written in arithmetic operation Sorry for my bad English.
I often see the following sentence

in polynomial time $O(n^2)$ arithmetic operation in $\mathbb{Z}$

or

in polynomial time $O(n^2)$ arithmetic operation in $\mathbb{F}_q$

Is my interpretation "arithmetic operation in $\mathbb{F}_q$" is correct?
i.e. for example a calculation $(5+3)\times (3-1)$ is $(5+3)\times (3-1)=8\times (3-1)=8\times 2=16$, so we have  3 arithmetic operation in $\mathbb{Z}$.
If the above interpretation is true, then can we count "addition" and "multiplication"?
I think there is big different at the point of time between "addition" and "multiplication".
Please help me.
 A: Depends on the Context. The writer has to specify which operations are "costly" & which are "easy" , then the given algorithm can be analysed to estimate the number of the costly operations & then claim $O(n^2)$ Etc.
There are Cases when "Comparison" is the Core Operation (eg Sorting) & those algorithms count that Operation to state that Eg Bubble Sort is $O(n^2)$ or other sorting algorithm is $O(n\log{n})$ Etc.
Such algorithms may have Multiplication & Addition involved , which are more costly yet do not much contribute to the total running time.
There are algorithms where "Comparison" is immaterial with Multiplication & Addition being the Concern. Eg Matrix Addition or Matrix Multiplication.
In that Case , the writer will specify that Multiplication & Addition are equally costly to then count them both.
Alternately , the writer will specify that Multiplication is very costly (we can ignore Addition) & thus count Multiplications.
Hence Depending on context , your Example has : either 3 Operations (1 Multiplication + 2 Additions) or 1 Operation (1 Multiplication) or 2 Operations (1 Multiplication + 1 Addition , ignoring the "Decrement Operation")
In Some cases , these are floating Point Numbers & all Calculations are somewhat "Equally Costly" , including "Comparison" , "Multiplication" , "Addition" , "Increment Operation" , "Decrement Operation" , "Power functions" , "trig functions" , "log functions" , Etc. Then we count all such Operations towards the total.
